Necklace rings and logarithmic functions

In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor Nr from the category of special λ-rings into the category of special λ-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper [Free Lie algebras and formal power series, J. Algebra 253(1) (2002) 167–188] to the case of graded Lie (super)algebras with a group action by applying the Euler–Poincaré principle.